scholarly journals Limit distribution of the quartet balance index for Aldous’s $(\beta \ge 0)$-model

2020 ◽  
Vol 47 (1) ◽  
pp. 29-44
Author(s):  
Krzysztof Bartoszek
Keyword(s):  
Limit Distribution ◽  
Balance Index

scholarly journals Exact and approximate limit behaviour of the Yule tree’s cophenetic index

2017 ◽  
Author(s):  
Krzysztof Bartoszek
Keyword(s):  
Limit Distribution ◽  
Heuristic Methods ◽  
Birth Process ◽  
Type Distribution ◽  
Continuous Branch ◽  
Branch Lengths ◽  
Limit Behaviour ◽  
Contraction Type ◽  
Approximate Limit ◽  
Balance Index

AbstractIn this work we study the limit distribution of an appropriately normalized cophenetic index of the pure–birth tree conditioned onncontemporary tips. We show that this normalized phylogenetic balance index is a submartingale that converges almost surely and inL2. We link our work with studies on trees without branch lengths and show that in this case the limit distribution is a contraction–type distribution, similar to the Quicksort limit distribution. In the continuous branch case we suggest approximations to the limit distribution. We propose heuristic methods of simulating from these distributions and it may be observed that these algorithms result in reasonable tails. Therefore, we propose a way based on the quantiles of the derived distributions for hypothesis testing, whether an observed phylogenetic tree is consistent with the pure–birth process. Simulating a sample by the proposed heuristics is rapid, while exact simulation (simulating the tree and then calculating the index) is a time–consuming procedure. We conduct a power study to investigate how well the cophenetic indices detect deviations from the Yule tree and apply the methodology to empirical phylogenies.

scholarly journals Limit distribution of the quartet balance index for Aldous’s β ≥ 0-model

2018 ◽  
Author(s):  
Krzysztof Bartoszek
Keyword(s):  
Fixed Point ◽  
Limit Distribution ◽  
Phylogenetic Trees ◽  
Contraction Operator ◽  
Balance Index

AbstractThis paper builds up on T. Martínez-Coronado, A. Mir, F. Rossello and G. Valiente’s work “A balance index for phylogenetic trees based on quartets”, introducing a new balance index for trees. We show here that this balance index, in the case of Aldous’s β ≥ 0-model, convergences weakly to a distribution that can be characterized as the fixed point of a contraction operator on a class of distributions.

3-VERTEX FULL BALANCE INDEX SET OF GRAPHS

2020 ◽  
Vol 9 (6) ◽  
pp. 3247-3264
Author(s):  
N. C. Devadas ◽  
H. J. Gowtham ◽  
S. D'Souza ◽  
P. G. Bhat
Keyword(s):  
Index Set ◽  
Balance Index

On the first zero of an empirical characteristic function

1991 ◽  
Vol 28 (3) ◽  
pp. 593-601 ◽  
Author(s):  
H. U. Bräker ◽  
J. Hüsler
Keyword(s):  
Characteristic Function ◽  
Limit Distribution ◽  
Random Variable ◽  
Empirical Characteristic Function ◽  
The Real ◽  
Exponential Decrease ◽  
Characteristic Process

We deal with the distribution of the first zero Rn of the real part of the empirical characteristic process related to a random variable X. Depending on the behaviour of the theoretical real part of the underlying characteristic function, cases with a slow exponential decrease to zero are considered. We derive the limit distribution of Rn in this case, which clarifies some recent results on Rn in relation to the behaviour of the characteristic function.

scholarly journals Non-local Solvable Birth–Death Processes

2021 ◽  
Author(s):  
Giacomo Ascione ◽  
Nikolai Leonenko ◽  
Enrica Pirozzi
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Limit Distribution ◽  
Spectral Decomposition ◽  
Strong Solutions ◽  
Stochastic Representation ◽  
Discrete Approximations ◽  
Local Difference ◽  
Non Local ◽  
Birth Death

AbstractIn this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth–death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth–death processes.

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